<h2>Problem 294</h2>
<div style="color:#666;font-size:80%;">29 May 2010</div><br />
<div class="problem_content">
<p>
For a positive integer k, define d(k) as the sum of the digits of k in its usual decimal representation.
Thus d(42) = 4+2 = 6.
</p>
<p>
For a positive integer n, define S(n) as the number of positive integers k < 10<img src="" style="display:none;" alt="^(" /><sup>n</sup><img src="" style="display:none;" alt=")" /> with the following properties :
<ul>
<li>k is divisible by 23 and
<li>d(k) = 23.
</ul>
</p>
You are given that S(9) = 263626 and S(42) = 6377168878570056.
</p>
<p>
Find S(11<img src="" style="display:none;" alt="^(" /><sup>12</sup><img src="" style="display:none;" alt=")" />) and give your answer mod 10<img src="" style="display:none;" alt="^(" /><sup>9</sup><img src="" style="display:none;" alt=")" />.
</p>

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